Galois coverings of algebras by locally support-finite categories

Galois coverings of algebras by l o c a l l y s u p p o r t - f i n i t e categories

Piotr Dowbor, Helmut Lenzing and Andrzej Skowroflski Throughout the paper

K w i l l denote a f i x e d a l g e b r a i c a l l y closed f i e l d . We w i l l

use f r e e l y d e f i n i t i o n s and notations introduced in [2]. Let K-category. We w i l l denote by

R be a l o c a l l y bounded

Mod R the category of a l l R-modules and by

mod R

(resp. ind R) i t s f u l l subcategory formed by a l l f i n i t e l y generated (resp. indecomposable f i n i t e l y generated) R-modules. For any R-module M we w i l l denote by supp M the support of objects a such that subcategory of such that finite

M, that i s , the f u l l subcategory of

M(a) • O. For each object

R formed by a l l objects of

a

of

R consisting of a l l

R, Ra

w i l l denote the f u l l

supp M, f o r a l l modules

M in

M(a) • O. Then f o l l o w i n g [3] R is l o c a l l y s u p p o r t - f i n i t e i f

(the number of objects is f i n i t e ) f o r each object

a

of

Ra

ind R is

R. Well-known

examples of l o c a l l y s u p p o r t - f i n i t e categories are f i n i t e and l o c a l l y representationf i n i t e categories. The main aim of this paper is to prove the theorem below. For the case of pr e s i d u a l l y f i n i t e groups this theorem was proved in [3]. Here, we give a new independent proof being a combination of Lemma I due to the second author and Lemma 2 stated in [3]. THEOREM. Let

R be a l o c a l l y s u p p o r t - f i n i t e K-category and l e t a group

K-automorphism of

generated R-modules. Then R/G

is l o c a l l y s u p p o r t - f i n i t e and the push-down functor

F~: Mod R ~ Mod R/G associated with the Galois covering b i j e c t i o n between the G-orbits of isoclasses of objects in of objects in

ind R/G. In p a r t i c u l a r

[ 4 , 5 , 6 ] ) i f and only i f LEMMA I. Let phisms of

R-modules. Let

R/G

F~ R ~ R/G

induces a

ind R and the isoclasses

R is representation-tame (in the sense of

is so.

R be a l o c a l l y bounded K-category and l e t a group

R act

G of

R act f r e e l y on the isoclasses of indecomposable f i n i t e l y

G of K-automor-

f r e e l y on the isoclasses of indecomposable f i n i t e l y generated

F~: Mod R ~ Mod R/G be the push-down functor and l e t

F.: Mod R/G ~Mod R be the pull-up functor associated with the Galois covering F: R ~ R/G. For a

module

X in

(i)

X ~ F~Y f o r some Y in

(ii)

F.X

(iii)

There is a f i n i t e l y

ind R/G the f o l l o w i n g conditions are equivalent ind R.

is a d i r e c t sum of f i n i t e l y generated R-modules. generated d i r e c t summand of

F.X.

92 Proof. ( i ) ~ ( i i ) . module

F.X ~ F.FAY ~ (iii)

If

~ (i).

is of the form

X

Assume t h a t

F.X

has a f i n i t e l y

F.X = Y ~ Z

is isomorphic to

where

Y

generated d i r e c t summand. Then is an object in

HomR/G(F~Y,X) ~ JR/G(FAY,X)

mod R/G. Consider a minimal l e f t

mod R. Then by [7, Theorem 3.6] morphism in

generated

R-

is obvious.

FAY. Since from [7, Lemma 3.5]

is enough to prove t h a t radical in

X ~ FAY f o r some f i n i t e l y

we have the required decomposition

@ gY. The i m p l i c a t i o n ( i i ) ~ ( i i i ) g C G

has a decomposition that

X

Y, then by [7, Lemma 3.2]

ind R. We shall show FAY

where

is indecomposable, i t

JR/G is the Jacobson

almost s p l i t morphism

F~f: F~Y ~ F~E

F.X

f : Y ~ E in

is a minimal l e f t almost s p l i t

mod R/G. Using f u n c t o r i a l p r o p e r t i e s of almost s p l i t morphisms [ I ] and

the f a c t t h a t

FA

is l e f t a d j o i n t to

F. we get the f o l l o w i n g commutative diagram

o f a b e l i a n groups with exact rows Hom(FAf,X)

I

i

Hom(f,F.X)

HomR(E,F.X)

~ HomR(Y,F.X)

Hence, f o r our aim, i t Indeed, i f

,/

~0 , HomR/G(F~Y,X)--, H°mR/G(F~Y'XJ/~R/G(FAY,X)/~

HomR/G(FAE,X)

it

I

~0

..... Coker Hom(f,F.X)

is enough to show t h a t

HomR(f,F.X)

is not an epimorphism.

is an epimorphism, then the canonical s p l i t monomorphism

f a c t o r i z e s through

f , and

assumption. Consequently For a f u l l

f

is also a s p l i t monomorphism, c o n t r a r y to our

HomR/G(F~Y,X) ~ JR/G(FAY,X)

subcategory

h: Y ~ F.X

and the Lemma is proved.

C of a l o c a l l y bounded K-category

R we w i l l

denote by

A

C the f u l l

subcategory of

R(b,a) ~ 0

f o r some b

LEMMA 2. Let

C be a f u l l

R formed by a l l objects

from

C. Then

Y

R(a,b) # 0

subcategory of a l o c a l l y bounded K-category

Res: Mod R ~ Mod C denote the r e s t r i c t i o n in

such t h a t

or

C.

A

there is a decomposition

a

Res X = Y @ Z

is an R-module and

in

R and l e t

f u n c t o r . Assume t h ) t f o r an R-module X ^ Mod C such t h a t supp Y is contained

× = Y ~ Z'

f o r some R-module

Z'

Proof. The f a c t t h a t Y is an R-module is a t r i v i a l consequence of the d e f i n i t i o n A--" of C. The required R-module Z' is constructed as f o l l o w s : For each object a E R A

we set in

Z ' ( a ) = Z(a)

R then

shows t h a t

if

Z ' ( a ) = Z(a) Z'

A

a E C and if

A

Z ' ( a ) = X(a)

~ C C and

is an R-module and

for

Z ' ( a ) = X(~)

× = Y @ Z'

in

a ¢ C. I f for

Mod R.

~ A

is a morphism

~ ¢ C. A simple

analysis

93

Proof o f the Theorem. From [7, Theorem 3.6] Fl: Mod R ~ Mod R/G

the push-down f u n c t o r

associated with the Galois covering

F: R ~ R/G

induces an

i n j e c t i o n of the required sets. In order to prove t h a t t h i s is also a s u r j e c t i o n we will

use Lemmas I and 2. Let

and l e t

a

Un, n 6 ~

be an o b j e c t of

X be an indecomposable f i n i t e l y

supp F.X. Consider the f o l l o w i n g i n f i n i t e

= { 0 , 1 , 2 . . . . } , of f i n i t e

is the f u l l

subcategory o f

generated

full

subcategories of

R defined as f o l l o w s : Uo

R/G-module

sequence

R defined as f o l l o w s : Uo

is the f u l l

subcategory of

R

A

formed by the o b j e c t finite

a

subcategory o f

Res: Mod R ~ Mod Un+I

and, f o r

R containing

functor

by our assumption composable f i n i t e l y Lemma I

Y

Y(a) + O. From [8, Chap X]

and f a i t h f u l

X

Un = Un_I . Then

a, is contained in some Un.

be the r e s t r i c t i o n

mod Un+I , Res(F.X)(a) ~ O, and l e t such t h a t

n > O, we set

f u n c t o r . Then

Res(F.X)

Ra, as a f u l l Let is an object o f

be an indecomposable d i r e c t summand o f the f u n c t o r

I : Mod Un+I ~ Mod R. Thus

Res

admits a l e f t a d j o i n t f u l l is an o b j e c t of

ind R and

supp Y c supp I(Y) c Ra c Un. Hence from Lemma 2, Y

is an inde-

generated d i r e c t summand of

is isomorphic to

I(Y)

Res(F.X)

F.X

in

Mod R. Consequently from

FAY and the theorem is proved.

References [I ] M. Auslander and I . Reiten, Representation theory of a r t i n algebras VI, Commun. Algebra 6 (1978), 257 - 300. [2 ] K. Bongartz and P. G a b r i e l , Covering spaces in r e p r e s e n t a t i o n theory, Invent. Math., 65 (1982), 331 - 378. [3 ] P. Dowbor and A. Skowro~ski, On Galois coverings of tame algebras, p r e p r i n t 1983, I - 12. [4 ] P. Dowbor and ~. Skowro~ski, On the r e p r e s e n t a t i o n type of l o c a l l y bounded c a t e g o r i e s , p r e p r i n t 1984, I - 7. [5 ] Ju. A. Drozd, On tame and w i l d matrix problems, In: Matrix problems, Kiev (1977), 104 - 114. [6 ] Ju. A. Drozd, Tame and w i l d m a t r i x problems, In: Representations and quadratic forms, Kiev (1979), 39 - 74. [7 ] P. G a b r i e l , The u n i v e r s a l cover of a r e p r e s e n t a t i o n - f i n i t e a l g e b r a , In: Representations o f Algebras, Springer Lecture Notes 903, 1981, 68 - 105. [8 ] S. Mac Lane, Categories f o r the working mathematician, S p r i n g e r - V e r l a g , 1971, 262 p.

P. Dowbor and A. Skowro~ski I n s t i t u t e o f Mathematics, Nicholas Copernicus U n i v e r s i t y Chopina 12/18 87-100 Toruh, Poland

Received

9.11.1984

H. Lenzing Fachbereich Mathematik der Universit~t-Gesamthochschule D-4790 Paderborn Germany